The approaches taken in [1]}, [2]}, [3]}, [4]}, [5]} are all based on classifying the contact vector fields, and then using an “integration” argument to obtain rigidity for contact mappings [6]}, [7]}. To carry out the latter step one has to know that the pushforward of a smooth contact vector field under a contact mapping is still a contact vector field (although possibly of low regularity, a priori). This pushforward assertion is obvious for \(C^2\)  diffeomorphisms and was shown to hold for Euclidean bilipschitz homeomorphisms in [5]}; this appears to be the minimal regularity needed to implement such an argument.
